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The paper claims that every non-degenerate neutral Schwarzschild–de Sitter black hole evaporates monotonically in the Unruh–de Sitter state, with empty de Sitter space as the only ultimate fate.

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2026-08-03 22:09 UTC pith:OI67ZNCX

load-bearing objection The core geometric result—κ_b > κ_c with a positive anomaly flux and a consistent GSL—is sound and worth publishing, but the abstract and Page-curve sections overstate what is actually derived. the 4 major comments →

arxiv 2511.11873 v3 pith:OI67ZNCX submitted 2025-11-14 hep-th gr-qc

The fate of Schwarzschild--de Sitter black holes: nonequilibrium evaporation

Damien A. Easson This is my paper
classification hep-th gr-qc MSC 83C5781T2083C80 PACS 04.70.Dy04.62.+v
keywords Schwarzschild–de Sitterblack hole evaporationtrace anomalyPolyakov actionUnruh–de Sitter stategeneralized second lawNariai limitentanglement islands
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper is trying to establish a definitive fate for Schwarzschild–de Sitter black holes—black holes sitting in a universe with a positive cosmological constant, bounded by both a black-hole horizon and a cosmological horizon. Using the two-dimensional trace anomaly of N conformal matter fields (the Polyakov action) in the spherically reduced dilaton theory, it derives a conserved heat flux J = (N/48π)(κ_b² − κ_c²) between the two horizons. The key geometric fact is that the black-hole surface gravity κ_b is always larger than the cosmological one κ_c inside the static patch, so the flux never changes sign: the black hole always loses mass and the only zero-flux equilibrium is the degenerate Nariai limit. The paper concludes that evaporation is monotonic, the generalized second law holds at every stage, and the endpoint is empty de Sitter space. A careful reader would care because this gives the first fully analytic, backreacted two-horizon evaporation solution in this setting.

Core claim

In the Unruh–de Sitter state, the Polyakov anomaly action yields a steady, radially conserved Killing flux J = (N/48π)(κ_b² − κ_c²), with state constants fixed by regularity on both future horizons. From the horizon identity 1 = (Λ/3)(r_b² + r_b r_c + r_c²), the paper proves κ_b − κ_c = (r_b + r_c)/2 (1/(r_b r_c) − Λ) > 0, so J > 0 and Ṁ = −J < 0 everywhere in the static patch: evaporation is monotonic. The flux also implies T_b Ṡ_b = −J and T_c Ṡ_c = +J, so Ṡ_gen > 0 and the generalized second law holds. The central claim: no finite-temperature equilibrium exists in neutral SdS; only the degenerate Nariai limit has zero flux, and the endpoint is empty de Sitter space.

What carries the argument

The load-bearing object is the two-horizon flux formula for the anomaly-induced Polyakov stress tensor, J = (N/48π)(κ_b² − κ_c²), evaluated in the Unruh–de Sitter state (state constants t_u = κ_b²/4, t_v = κ_c²/4). The argument turns on the purely geometric inequality κ_b − κ_c = (r_b + r_c)/2 (1/(r_b r_c) − Λ) > 0, which follows from the SdS horizon relation and guarantees the flux sign for every allowed (M, Λ). This identity, joined to the adiabatic Eddington–Finkelstein mass-balance law Ṁ = −J, carries the derivation from flux to entropy production and to the thermodynamic Page-curve estimate.

Load-bearing premise

The load-bearing premise is that the physical vacuum is the Unruh–de Sitter state, defined by regularity on both future horizons; if a different vacuum governs real SdS black holes, the flux direction—and with it the monotonic-evaporation fate—can change.

What would settle it

A four-dimensional computation of ⟨T_μν⟩ in the Unruh–de Sitter state on fixed SdS that found an inward (sign-reversed) conserved Killing flux near the black-hole horizon would falsify the monotonic-evaporation claim; so would an explicit regular neutral-SdS solution with κ_b = κ_c ≠ 0.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • Neutral SdS black holes never anti-evaporate: the flux direction is fixed by geometry, so the mass decreases monotonically for all allowed (M, Λ).
  • The generalized second law holds throughout evaporation: the cosmological horizon's entropy gain exceeds the black hole's loss, with Ṡ_gen > 0 until the Nariai limit.
  • The only equilibrium of the system is the Nariai configuration, where the horizons coincide and the surface gravities vanish; no lukewarm finite-temperature state exists for neutral SdS.
  • The same steady flux fixes the growth of the radiation entropy, producing a thermodynamic (non-microscopic) Page time and an island-dominated turnover consistent with unitarity.
  • As M → 0 the mass-loss law reproduces the familiar M³ evaporation timescale of four-dimensional Schwarzschild black holes, with the cosmological-horizon temperature approaching √(Λ/3).

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Because the sign structure of Δ depends only on horizon geometry, not on the details of the 2D reduction, the monotonic-evaporation conclusion should survive in four dimensions with only the evaporation rate rescaled—a testable quantitative extension.
  • The flux formula is identical to the heat current of a 1+1 CFT between two reservoirs, so the static patch can be viewed as a finite thermal cavity; this suggests checking the entropy-production inequality in laboratory analogues of moving mirrors or optical horizon analogues.
  • The decisive physical input is the vacuum choice: if a four-dimensional calculation picks a different state, the flux direction could flip near the Nariai limit, so the framework functions as a sharp diagnostic—measure the renormalized stress tensor in the Unruh–de Sitter state and compare the flux sign.
  • The Page-curve part is deliberately an estimate, not an extremization; a follow-up using the explicit Polyakov field to extremize S_gen in the backreacted geometry would convert the qualitative turnover into a precise Page-time prediction.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper constructs a two-dimensional dilaton-gravity model from spherical reduction of 4D Schwarzschild–de Sitter, adds the Polyakov trace-anomaly action for N conformal fields, and analyzes the semiclassical backreaction in the Unruh–de Sitter state. It derives a conserved Killing-energy flux J = N/(48π)(κ_b² − κ_c²), the mass-loss law Ṁ = −J, and proves via horizon identities that κ_b > κ_c throughout the static patch, implying monotonic evaporation toward empty de Sitter with the Nariai geometry as the only zero-flux limit. The paper further derives local Tolman temperatures, a first-law identity, the generalized second law, and a thermo-controlled Page-curve estimate based on min{S_rad, S_b}. The central algebraic core—Eqs. (32)–(34), the mass-loss law, and the entropy-production inequality—is internally consistent. The main weaknesses are the state-dependence of the evaporation direction, an apparent normalization ambiguity in the flux derivation, the adiabatic rather than fully-backreacted nature of the solution, and the heuristic status of the island/Page-curve section.

Significance. If accepted, the model provides a rare fully analytic two-horizon evaporation description with no fitted parameters beyond N, Λ, and M₀. The explicit inequalities κ_b ± κ_c, the first-law identity 2δM = T_b δS_b − T_c δS_c, and the GSL rate J(1/T_c − 1/T_b) > 0 are clean, checkable results that could serve as a useful benchmark for multi-horizon semiclassical thermodynamics. The Page-curve construction is suggestive but not a derivation of quantum extremal surfaces. The central physical claim—that every nondegenerate neutral SdS black hole evaporates monotonically—is conditional on the UdS state choice and on the 2D-to-4D sign robustness, neither of which is proven from the 4D theory. These caveats do not invalidate the model, but they need to be stated with much more discipline before the paper can claim to describe the actual fate of SdS black holes.

major comments (4)
  1. [§III.A–III.B, Eqs. (16), (21)] There is a normalization inconsistency in the flux formula. Eq. (16) defines ⟨T_uu⟩ = −N/(12π)[...] + t_u(u) and ⟨T_vv⟩ = −N/(12π)[...] + t_v(v), with t_u, t_v carrying no explicit N factor. Regularity is said to fix t_u = κ_b²/4, t_v = κ_c²/4. Then the state-dependent contribution to T_vv − T_uu is t_v − t_u = −(κ_b² − κ_c²)/4, which gives a Killing flux proportional to (κ_b² − κ_c²)/4, not N/(48π)(κ_b² − κ_c²). The later relation in §V.B, J = (N/48π)(t_+ − t_−), is only consistent if t_± implicitly contain an N/(12π) prefactor, but this is never defined. This affects all quantitative time scales, including τ_evap in Eq. (27) and t_Page in Eq. (60). The sign would survive, but the claimed quantitative mass-loss law needs a clear convention fix.
  2. [§III.B, §VI, and [24]] The evaporation direction is not a geometric theorem about 4D SdS; it follows only after selecting the Unruh–de Sitter state with t_u = κ_b²/4, t_v = κ_c²/4. The paper itself lists alternative states where the direction changes or flux vanishes, and the cited Bousso–Hawking analysis [24] found anti-evaporation near Nariai in a different state. The abstract and conclusions state that SdS black holes evaporate monotonically, without the essential qualifier 'in the UdS state of the 2D model.' Appendix A.5 only asserts that dimensional-reduction and greybody effects rescale J, not that the 4D flux direction is controlled by the 2D κ_b > κ_c inequality. Please either justify UdS as the physical state for 4D neutral SdS, or restrict the central claims to the reduced model and explicitly discuss the state-dependence and its conflict with [24].
  3. [§V, Appendices A–B] The phrase 'fully backreacted solution' overstates what is actually derived. Appendix A proves that no exact static solution with nonzero flux exists, and Appendix B derives Ṁ = −J only at leading adiabatic order, neglecting ∂_v f and ∂_v² ψ. Section V describes the result as 'closed form up to quadrature' and 'adiabatic (quasi-stationary).' No explicit closed-form backreacted metric with M(v), the Polyakov field, and the UdS stress tensor is presented. The abstract's claim of 'a fully analytic, backreacted solution' and 'exactly solvable model' should be tempered to 'a leading-order adiabatic, quasi-stationary solution.' This distinction matters because several later statements about horizon regularity and QES rely on the status of the backreacted geometry.
  4. [§IX.C, Eqs. (55)–(60)] The Page-curve construction is imposed rather than derived. The unitary entropy is defined by S_rad^unitary = min{S_rad, S_b}, and the Page condition S_rad(t_Page) ≃ S_b(t_Page) is assumed, not obtained from extremizing S_gen as prescribed in Eq. (51). The claim that 'the island endpoint lies exponentially close to the black-hole horizon' is asserted without a QES calculation. The text itself says a full QES analysis is left for future work. In view of this, the abstract's statement that the framework 'shows how quantum extremal surfaces and entanglement islands emerge naturally' is too strong. This section is best presented as a heuristic, thermo-controlled estimate, not as a derivation of islands in the backreacted SdS geometry.
minor comments (5)
  1. [§V.B, footnote 4] The notation t_± is introduced without a definition linking it to t_u, t_v of Eq. (16). If t_+ = t_u and t_− = t_v, then J = (N/48π)(t_+ − t_−) conflicts with the stated UdS values; please define the precise normalization.
  2. [§III.B, Eq. (24)] The locution 't_u/v = κ_h²/4' is typographically ambiguous; write t_u = κ_b²/4 and t_v = κ_c²/4 separately.
  3. [§IV, Eq. (27)] Specify whether M in τ_evap ∼ M/|Ṁ| is the 2D Casimir parameter or the 4D physical mass. Earlier, M_phys = M/(2G₄) is defined; using the same symbol for both in time-scale estimates is confusing.
  4. [§IX.C, Fig. 3] The caption says 'no additional fit parameters are introduced,' but the Page curve depends on the choice of the island condition S_rad ≃ S_b. It would be clearer to call this a 'model assumption' rather than a parameter-free prediction.
  5. [References] Given the centrality of the state-dependence issue, reference [24] should be discussed in more detail in the main text, not only listed; the reader needs to know why the UdS state is preferred over the Hartle–Hawking-like state of [24] for describing SdS evaporation.

Circularity Check

1 steps flagged

Evaporation direction and GSL are independent; the Page/unitarity claim is imposed by the min{S_rad,S_b} ansatz.

specific steps
  1. self definitional [Section IX.C 'A thermo-controlled Page-curve estimate' and 'Refined Page condition', Eq. (56) and Fig. 3 caption]
    "At the semiclassical level this equality is well approximated by two standard assumptions: (i) the no–island saddle is dominated by the coarse–grained radiation entropy S_rad, and (ii) the island saddle is dominated by the black hole’s Bekenstein–Hawking entropy S_b. Under these approximations, the Page condition simplifies to S_rad(t_Page)≃S_b(t_Page). ... S(unitary)_rad(t) = min{S_rad(t), S_b(t)}."

    The 'unitary (island)' entropy curve is defined as min{S_rad,S_b}, and the Page time is defined by S_rad(t_Page)≃S_b(t_Page). Therefore the Page turnover and the claimed 'restoration of unitarity' are consequences of the imposed ansatz, not outputs of the anomaly-induced calculation. The paper itself defers the actual QES extremization ('A detailed QES extremization in the fully backreacted SdS geometry is left for future work') and later says the construction 'remains an effective resolution ... it does not yet reveal the microscopic mechanism.' Thus the information-recovery 'prediction' reduces by construction to the min formula.

full rationale

The load-bearing evaporation chain is not circular. Eq. (34), κ_b−κ_c = (r_b+r_c)/2 (1/(r_b r_c)−Λ) > 0, is derived algebraically from the SdS horizon relation (32); the mass-loss law Ṁ=−J follows from the covariant Casimir balance in Appendix B; and the GSL sign is a direct consequence of the first-law/Clausius relations (46)–(48). None of these steps uses a fitted parameter or a load-bearing self-citation. The flux formula (21) is imported as the standard 2D CFT Unruh flux, with state constants t_u=κ_b²/4, t_v=κ_c²/4 fixed by horizon regularity; that is an explicit state assumption (UdS), not a circular fit, and Appendix A.5 transparently assumes that dimensional-reduction and greybody effects only rescale J without changing its sign. The genuine circularity is confined to Sec. IX: the 'unitary' Page curve is defined as min{S_rad,S_b} and the Page time is defined by S_rad(t_Page)=S_b(t_Page), so 'restoration of unitarity' is an input ansatz rather than a derived result, as the paper partially concedes by deferring QES extremization and calling the construction an effective resolution. Self-citations in the paper ([19]–[21], [23], [48]) are peripheral and not load-bearing. Accordingly, the central evaporation and GSL results are independent, while the information-flow claim has a definitional circularity; overall score 4. The possible factor-of-4 normalization issue between Eq. (16)/note 4 and Eq. (21) is a consistency/correctness concern, not a circularity.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 0 invented entities

No new particles, forces, or dimensions are introduced. The ledger burden is concentrated in the state choice (UdS), the 2D-to-4D fidelity assumption (App. A.5), the imported CFT flux formula, and the island prescription used for the Page estimate. No numbers are fitted to external data; N, Λ, and M₀ are physical inputs, and the unspecified Fig. 3 parameters are a minor reproducibility gap.

free parameters (3)
  • N (number of conformal matter fields)
    Central charge of the matter sector; sets the magnitude of the flux J=(N/48π)(κ_b²−κ_c²) and the Page time (Eqs. 21, 60). Physical input, not fitted, but all quantitative mass-loss and entropy-production results scale with it.
  • Λ (cosmological constant)
    Physical input fixing the Nariai mass M_N=1/(3√Λ) and the static-patch domain 0<9M²Λ<1. Not fitted to data.
  • M₀ (initial black-hole mass)
    Boundary condition for the mass-evolution trajectory. Fig. 3 uses unspecified 'representative' values, which prevents exact reproduction of the Page-curve figure.
axioms (6)
  • domain assumption The physical state for an evaporating SdS black hole is the Unruh–de Sitter state, fixed by regularity on both future horizons (t_u=κ_b²/4, t_v=κ_c²/4).
    Entered in §III.B and used for Eq. (21). If the physical state differs (e.g., Hartle–Hawking-like near Nariai), the flux direction and magnitude change; the paper's own cited literature [24] found anti-evaporation in a different treatment. This is the load-bearing physics input.
  • domain assumption The 2D Polyakov action with central charge c=N captures the complete semiclassical s-wave backreaction of the 4D SdS static patch, up to an overall rescaling of J.
    Invoked in §II–III and App. A.5, which admits the dimensional-reduction anomaly and greybody factors modify the flux magnitude; the sign structure is argued to be robust but not proven from 4D.
  • standard math Standard CFT steady-state flux formula J=(πc/12)(T_b²−T_c²) applies to the two-horizon system.
    Used in Eq. (21)/(49); this is the textbook 1+1 CFT heat-flow result, quoted rather than derived within the paper.
  • domain assumption Adiabatic approximation |Ṁ|≪κ_b M keeps the geometry quasi-static at each instant.
    Stated in §III.A (footnote 3); required for treating the flux as radially constant while M evolves (Eq. 20). The paper notes this breaks down near the Planck scale.
  • domain assumption Island prescription: the Page turnover is obtained from S_rad(t_Page)=S_b(t_Page) and S_unitary=min{S_rad,S_b}.
    Assumptions (i)-(ii) in §IX.C.1, Eq. (56). The turnover is imposed by this prescription rather than derived from a QES extremization, which the paper defers to future work (§I).
  • domain assumption GSL is quantified by horizon Clausius relations T_b Ṡ_b=−J, T_c Ṡ_c=+J.
    §VIII.A.1, Eq. (46). Assumes the conserved flux J is the heat current absorbed/emitted by each horizon; consistent bookkeeping but an input to the derivation.

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Cite this review

Pith. "Pith review of The fate of Schwarzschild--de Sitter black holes: nonequilibrium evaporation." pith.science (2026). https://pith.science/paper/OI67ZNCX

@misc{pith2026251111873,
  author       = {Pith},
  title        = {Pith review of: The fate of Schwarzschild--de Sitter black holes: nonequilibrium evaporation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OI67ZNCX}},
  note         = {Machine review of arXiv:2511.11873}
}
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read the original abstract

We present a fully analytic treatment of Schwarzschild--de~Sitter (SdS) black-hole evaporation in two-dimensional dilaton gravity with anomaly-induced backreaction. Starting from the spherical reduction of four-dimensional Einstein gravity with a cosmological constant, we construct an exactly solvable 2D model that captures the full causal and thermodynamic structure of the SdS static patch, including both black-hole and cosmological horizons. Incorporating the trace anomaly of $N$ conformal matter fields via the Polyakov action, we determine the evolution of the black-hole mass and geometry in the Unruh--de~Sitter state, track the steady nonequilibrium Hawking flux, and compute local thermodynamic observables for static observers. The conserved Killing energy flux drives an irreversible heat current from the black hole to the cosmological horizon whenever their surface gravities differ, ensuring monotonic entropy growth and satisfaction of the generalized second law. We prove that $\kappa_b>\kappa_c$ throughout the physical static patch, so the only zero-flux configuration is the Nariai limit where the horizons coincide. Extending the framework to the quantum-information regime, we construct a thermo-controlled estimate of the Page curve and show how quantum extremal surfaces and entanglement islands emerge naturally within the anomaly-induced steady state. These results constitute a fully analytic, backreacted solution for SdS evaporation that unifies semiclassical thermodynamics and information flow in a cosmological setting, thereby elucidating the ultimate fate of evaporating black holes in de~Sitter space.

Figures

Figures reproduced from arXiv: 2511.11873 by Damien A. Easson.

Figure 1
Figure 1. Figure 1: FIG. 1: Portion of the Penrose diagram for Schwarzschild–de Sitter emphasizing the static [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Evaporation phase diagram for Schwarzschild–de Sitter black holes. Each point [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Thermo-controlled Page curve for Schwarzschild–de Sitter. The solid (yellow) [PITH_FULL_IMAGE:figures/full_fig_p030_3.png] view at source ↗

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Reference graph

Works this paper leans on

79 extracted references · 44 linked inside Pith · cited by 3 Pith papers

  1. [1]

    Flux balance and horizon Clausius relations Energy conservation implies a single radially conserved flux of Killing energy across the patch. DefiningJ>0 as the outward (increasing-r) flux measured by static observers, each horizon obeys its own local Clausius relation, Tb ˙Sb =− J, T c ˙Sc = +J,(46) expressing that the black hole loses heat while the cosm...

  2. [2]

    quantum expansions

    Generalized second law Combining (46) gives the total gravitational entropy production rate, ˙Sgen = ˙Sb + ˙Sc =J 1 Tc − 1 Tb >0 (κ b > κc),(48) 6 Different conventions exist for defining temperatures and entropies in 2D dilaton gravity. Some authors define the Wald entropy asS= 2πX, which leads to a first law of the formδM= 2T δS, whereas in the spherica...

  3. [3]

    The global state of the full SdS spacetime is pure, yet a static observer has access only to the inter-horizon region

    Entanglement structure In the de Sitter static patch, the radiation field is not entangled solely with the black-hole interior (as in asymptotically flat evaporation) but also with degrees of freedom associated with the cosmological horizon. The global state of the full SdS spacetime is pure, yet a static observer has access only to the inter-horizon regi...

  4. [4]

    In theUnruh–de Sitterstate considered here, the steady flux is directed outward from the black–hole horizon toward the cosmological horizon

    Cosmological counterpart By symmetry, a complementary quantum extremal surface (QES) can lie just inside the cosmological horizonr c, corresponding to an island for observers whose algebra is anchored outside the static patch, in the neighboring de Sitter region. In theUnruh–de Sitterstate considered here, the steady flux is directed outward from the blac...

  5. [5]

    Refined Page condition The Page timet Page is defined as the moment when the generalized entropies of the two competing saddles become equal, S(no–island) gen (tPage) =S (island) gen (tPage). At the semiclassical level this equality is well approximated by two standard assumptions: (i) the no–island saddle is dominated by the coarse–grained radiation entr...

  6. [6]

    In the static gauge one has √−g= 1 and R=−f ′′(r),□Φ =∂ r fΦ ′(r) for any scalar Φ(r).(A6) Notation.A prime onforψdenotes∂ r, while a prime on ˜Vdenotesd/dX

    Preliminaries in static gauge We work with the localized action S= 1 4G4 Z √−g XR+ 2 ˜V(X) − N 96π Z √−g (∇ψ)2 + 2ψR , (A2) whose variation yields (withT (state) µν the conserved, traceless state contribution) R+ 2 ˜V ′(X) = 0,(A3) 1 4G4 −g µν□X+∇ µ∇νX− ˜V(X)g µν =T (ψ) µν +T (state) µν ,(A4) □ψ=R, T (ψ) µν =− N 24π ∇µ∇νψ−g µν□ψ− 1 2 ∇µψ∇νψ+ 1 4 gµν(∇ψ)2 ...

  7. [7]

    First integrals From (A5) and (A6), (f ψ′)′ =R=−f ′′(r)⇒f ψ ′ =−f ′ +C ψ,(A7) whereC ψ is an integration constant fixed by horizon regularity (in particularC ψ = 0 for regular future horizons). Taking the trace of (A4) and using tracelessT (state) together with (A3) gives 1 4G4 −□X−2 ˜V(X) = N 24π R=− N 12π ˜V ′(X),(A8) which in the gauge (A1) yields the ...

  8. [8]

    Static momentum constraint and a no-go for exact static flux The (r, t) component of Eq. (A4) reads 1 4G4 grt□X− ∇r∇tX− ˜V(X)g rt =T (ψ) rt +T (state) rt .(A12) 36 In a static metric ansatz withg rt = 0 andX=X(r), the mixed derivative∇ r∇tXvanishes, so the left-hand side of (A12) is identically zero. The off–diagonal field equation then enforces T (ψ) rt ...

  9. [9]

    Quadratures and horizon regularity (static, zero-flux case) In the strictly static caseJ= 0, regularity onbothfuture horizons corresponds to a Hartle–Hawking–dS–type configuration, which for neutral SdS exists only at the Nariai limit. By contrast, the Unruh–dS state (regular on the future horizons but carrying nonzero flux) requires a time-dependent ansa...

  10. [10]

    These corrections can modify the quantitative relation between the 4D and 2D fluxes, rescaling the magnitude ofJor shifting the ef- fective dilaton potential

    Dimensional-reduction anomaly The present analysis employs the standard spherically reduced two–dimensional dila- ton–gravity action augmented by the Polyakov term, which correctly captures the structure of the trace anomaly ofNconformal matter fields but omits the so–calleddimensional- reduction anomaly: additional state-dependent terms that arise when i...

  11. [11]

    Comment on coordinates—In the EF chart the metric is non-diagonal, so its inverse has bothg rv = 1 andg rr =f(v, r)

    EF ansatz and basic identities To describe flux solutions it is convenient to work in advanced Eddington–Finkelstein (EF) coordinates (v, r) with ds2 =−f(v, r)dv 2 + 2dv dr, X=X(r) (for the Schwarzschild–de Sitter reduction,X=r 2), ψ=ψ(v, r),(B2) where the classical metric function is fcl(M,Λ;r) = 1− 2M r − Λ 3 r2, and we allow the mass parameterM=M(v) to...

  12. [12]

    Mixed Einstein equation and state flux From the localized action (see Appendix A), the (v, r) component of the semiclassical Einstein equation is 1 4G4 h 1 2 f ′X ′ −∂ r f X′ − ˜V(X) i =T (ψ) vr +T (state) vr ,(B7) where we have used∇ v∇rX= 1 2 f ′X ′,g vr = 1, and□X=∂ r(f X′). Taking∂ v of (B7), and noting thatX ′(r) and ˜V(X) arev-independent, yields 1 ...

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